Anything is possible, you just have to believe. – We Solve Problems

#### Anything is possible, you just have to believe.

While solving a mathematical problem, it might be that we don’t have enough of an existing understanding to deal with it. Sometime we can’t believe a given situation (a described agreement) is possible simply because of our lack of experience in that particular area of maths. In those cases we have to push ourselves forward to expand our learning. It’s important not to give up; the only way you learn is by comparing what you don’t know with what you do until you also know that. You have to work through all the examples, despite the fact you are trying to prove something that you think is actually incorrect, until you reach a definite conclusion one way or another. Unless you have rigorously worked through every possible consideration you cannot say for sure that it wasn’t possible. Sochiro Honda, of Honda Motors, said ”Success is 99% failure” – so giving up is not an option!

Problems:

#### Dissections, partitions, covers and tilings , Plane dissected by lines

Jennifer draws a hexagon, and a line passing through two of its vertices. It turns out one of the figures in which the original hexagon is divided is a heptagon. Show an example of a hexagon and a line for which it is true.

#### Algebra and arithmetics , Arithmetic operations. Number identities , Prime numbers

There are four numbers written in a row. The first number is 100. It is known that the first number if divided by the second number is a prime number, the second number if divided by the third number is a prime number, and the third number if divided by the fourth number is also a prime number. Can all these prime numbers be distinct?

#### Algebra and arithmetics , Arithmetic operations. Number identities

George claims that he knows two numbers such that their quotient is equal to their product. Can we believe him? Prove him wrong or provide a suitable example.

#### Dissections, partitions, covers and tilings , Plane dissected by lines

Can Jennifer draw an octagon and a line passing through two of its vertices in such a way that this line cuts a 10-gon from it?

#### Algebra and arithmetics , Arithmetic operations. Number identities , Prime numbers

In the context of Example 14.2 what is the answer if we have five numbers instead of four? (i.e., can we get four distinct prime numbers then?)

#### Algebra and arithmetics , Arithmetic operations. Number identities

Now George is sure he found two numbers with the quotient equal to their sum. And on top of that their product is still equal to the same value. Can it be true?

#### Area of a circle, sector, segment , Circles , Plane geometry

A maths teacher draws a number of circles on a piece of paper. When she shows this piece of paper to the young mathematician, he claims he can see only five circles. The maths teacher agrees. But when she shows the same piece of paper to another young mathematician, he says that there are exactly eight circles. The teacher confirms that this answer is also correct. How is that possible and how many circles did she originally draw on that piece of paper?

#### Logic and set theory , Set theory and logic

A group of three smugglers is offered to smuggle a chest full of treasures across the dangerous river. The boat they possess is old and frail. It can carry three smugglers without the chest, or it can carry the chest and only two smugglers. The price for this job is extremely high, and the gang is more than interested in completing the job. Think of a strategy the smugglers should follow to successfully transit the chest and themselves to the other shore.

#### Constructions

It is easy to construct one equilateral triangle from three identical matches. Can we make four equilateral triangles by adding just three more matches identical to the original ones?

#### Dissections, partitions, covers and tilings , Plane dissected by lines

(a) Can one fit 4 letters “T” (see the picture below) in a $6\times6$ square box?

We do not allow any overlappings to occur.$\\$
(b) Can we fit them in a square with smaller side length?

#### Dissections (other) , Dissections, partitions, covers and tilings

After having lots of practice with cutting different hexagons with a single cut Jennifer thinks she found a special one. She found a hexagon which cannot be cut into two quadrilaterals. Provide an example of such a hexagon.

#### Logic and set theory , Set theory and logic

My mum once told me the following story: she was walking home late at night after sitting in the pub with her friends. She was then surrounded by a group of unfriendly looking people. They demanded: “money or your life?!” She was forced to give them her purse. She valued her life more, since she was pregnant with me at that time. According to her story she gave them two purses and two coins. Moreover, she claimed that one purse contained twice as many coins as the other purse. Immediately, I thought that the mum must have made a mistake or could not recall the details because of the shock and the amount of time that passed after that moment. But then I figured out how this could be possible. Can you?

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